Open filters and measurable cardinals.

In this paper, we investigate the poset OF ( X ) of free open filters on a given space X. In particular, we characterize spaces for which OF ( X ) is a lattice. For each n ∈ N we construct a scattered space X such that OF ( X ) is order isomorphic to the n-element chain, which implies the affirmative answer to two questions of Mooney. Assuming CH we construct a scattered space X such that OF ( X ) is order isomorphic to ( ω + 1 , ≥ ) . To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of β ( κ ) . Assuming the existence of n measurable cardinals, for every m 0 , … , m n ∈ N we construct a space X such that OF ( X ) is order isomorphic to ∏ i = 0 n m i . Also, we show that the existence of a metric space possessing a free ω 1 -complete closed, G δ , F σ or Borel ultrafilter is equivalent to the existence of a measurable cardinal.